Systematics of flux tubes in the dual Ginzburg-Landau theory and Casimir scaling hypothesis: folklore and lattice facts

The ratios between string tensions $\sigma_D$ of color-electric flux tubes in higher and fundamental SU(3) representations, $d_{D} \equiv \sigma_{D}/\sigma_{\bf 3}$, are systematically studied in a Weyl symmetric formulation of the DGL theory. The ratio is found to depend on the Ginzburg-Landau (GL) parameter, $\kappa \equiv m_{\chi}/m_{B}$, the mass ratio between monopoles ($m_{\chi}$) and dual gauge bosons ($m_{B}$). While the ratios $d_{D}$ follow a simple flux counting rule in the Bogomol'nyi limit, $\kappa=1.0$, systematic deviations appear with increasing $\kappa$ due to interactions between fundamental flux inside a higher representation flux tube. We find that in a type-II dual superconducting vacuum near $\kappa = 3.0$ this leads to a consistent description of the ratios $d_{D}$ observed in lattice QCD simulations.